\(\,\)
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The diagram shows
two particles, \(P\) and \(Q\),
connected by a light inextensible string which passes over a
smooth block fixed to a horizontal table.
The
cross-section of the block is a quarter circle with centre \(O\), which
is at the edge of the table, and radius \(a\). The angle between
\(OP\) and the table is \(\theta\).
The masses of \(P\) and \(Q\) are \(m\) and \(M \), respectively,
where \(m < M\).
Initially, \(P\) is held at rest on the table and in contact with the block,
\(Q\) is
vertically above \(O\), and the string is taut.
Then \(P\) is released. Given that, in the subsequent motion,
\(P\)
remains in contact with the block as \(\theta\)
increases from \(0\) to \(\frac12\pi\),
find an expression, in terms of \(m\), \(M\), \(\theta\) and \(g\),
for the normal reaction of the block on \(P\) and show
that
\[
\frac{m}{M} \ge \frac{\pi-1}3\,.
\]